1.5A

Systems of 2 Linear Equations

  • A system of two linear equations in variables $x$ and $y$: $$\begin{cases} a_1 x + b_1 y = d_1 \\ a_2 x + b_2 y = d_2 \end{cases}$$
  • Three Possible Outcomes:
    • Unique Solution: The lines intersect at a single point $(x, y)$. For nonvertical lines, the slopes are different ($m_1 \ne m_2$).
    • No Solution (Inconsistent): The lines are parallel with different intercepts ($m_1 = m_2, c_1 \ne c_2$).
    • Infinitely Many Solutions (Dependent): The equations represent the identical line ($m_1 = m_2, c_1 = c_2$).
1.5B

Systems of 3 Linear Equations in 3 Variables

  • In real-world applications, three unknown quantities ($x, y, z$) require three independent linear relationships for a unique solution: $$\begin{cases} a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \end{cases}$$
  • Using GDC:
    • TI-84 Plus: Use Apps → PlySmlt2 → Simultaneous Eqn Solver with 3 equations and 3 unknowns.
    • TI-Nspire: Use Menu → Algebra → Solve System of Linear Equations.
    • Casio fx-CG50: Use Equation Mode (EQUA) → Simultaneous (F1) → 3 Unknowns (F2).

Elimination, classification and validation

  • Elimination: multiply entire equations to make one variable’s coefficients equal, subtract, solve the reduced equation, then substitute back. Example $2x+y=10$, $x-y=2$ gives $3x=12$, x4,y2.
  • Two-variable determinant: $a_1b_2-a_2b_1\ne0$ gives one solution. If zero, eliminate and distinguish $0=0$ from $0=c\ne0$. The slope description applies only to nonvertical lines.
  • Worked classifications: $x+y=3,\;2x+2y=6$ gives $(x,y)=(t,3-t)$ for any real t. Replacing6by7 gives no solution.
  • Three different equations need not be independent. Three planes can intersect in a point or a line, coincide, or have no common point. A singular coefficient matrix does not itself distinguish no solution from infinitely many.
  • For calculator input keep the variable order fixed and enter0 for missing coefficients. Check every original equation. A solver error requires classification; it is not proof of no solution.
  • Define quantities and units before forming equations. Nonnegative weights and whole-number counts are contextual constraints; a unique real solution can still be infeasible.

10 Practice Exercises

Attempt each problem before opening the solution.

Exercise 1 (Paper 1 — 3 Marks): A cinema charges $\$12$ for an adult ticket and $\$8$ for a child ticket. A group of 14 people spends a total of $\$136$. Find the number of adult tickets and child tickets purchased.
Let $a$ = adult tickets, $c$ = child tickets.
Equation 1: $a + c = 14$
Equation 2: $12a + 8c = 136$ [M1]
From Eq 1, $c = 14 - a$. Substitute into Eq 2:
$12a + 8(14 - a) = 136 \implies 12a + 112 - 8a = 136 \implies 4a = 24 \implies a = 6$ [A1]
$c = 14 - 6 = 8$ [A1]

Answer: 6 adult tickets, 8 child tickets

Exercise 2 (Paper 2 — 5 Marks): A bakery sells three types of cookies: Chocolate ($x$), Oatmeal ($y$), and Peanut Butter ($z$).
  • The price of 2 Chocolate, 3 Oatmeal, and 1 Peanut Butter cookie is $\$11.00$.
  • The price of 3 Chocolate, 1 Oatmeal, and 2 Peanut Butter cookies is $\$10.50$.
  • The price of 1 Chocolate, 4 Oatmeal, and 3 Peanut Butter cookies is $\$15.50$.
  1. Write down a system of 3 linear equations representing this information.
  2. Use your GDC to find the individual price of each type of cookie.
(a) System of equations: $$\begin{cases} 2x + 3y + z = 11.00 \\ 3x + y + 2z = 10.50 \\ x + 4y + 3z = 15.50 \end{cases}$$ [M1][A1]
(b) Enter coefficients into GDC matrix / polynomial simultaneous solver [M1]:
$x = \$1.50$ [A1]
$y = \$2.00$ [A1]
$z = \$2.00$ [A1]
Check: $2(1.5)+3(2)+2=11$, $3(1.5)+2+2(2)=10.5$, and $1.5+4(2)+3(2)=15.5$.

Answer: Chocolate = $\$1.50$, Oatmeal = $\$2.00$, Peanut Butter = $\$2.00$

Exercise 3

Solve $3x+2y=16$, $x-y=2$ by elimination.

Twice the second equation added to the first gives5x20; x4,y2. Checks:12+4=16 and4−2=2.

Exercise 4

Classify $x+2y=5$, $2x+4y=10$ and describe every solution.

The second equation is twice the first. Infinitely many: $(x,y)=(5-2t,t)$, t real.

Exercise 5

Classify $x+2y=5$, $2x+4y=11$.

Subtract twice the first from the second to get0=1. No solution; the lines are parallel and distinct.

Exercise 6

Solve $x=3$, $2x+y=8$ and explain why slopes are unnecessary.

Substitute x3 to obtainy2. The vertical line and oblique line meet at(3,2).

Exercise 7

Solve $x+y+z=6$, $x-y+z=2$, $2x+y-z=1$.

Subtract first two to gety2. Then x+z4 and2x−z=−1, sox1,z3. Check each equation:6,2,1.

Exercise 8

Find all solutions of $x+y+z=6$, $2x+2y+2z=12$, $x-y=0$.

x=y=t,z=6−2t. The first two equations are dependent. If all quantities are nonnegative,0≤t≤3.

Exercise 9

A 10kg mixture uses ingredients costing2/kg and5/kg. The total cost is32. Find both masses.

Let x,ykg: x+y10,2x+5y32. Subtract2(x+y) to get3y12, so y4,x6. Units and nonnegativity are valid.

Exercise 10

A box contains10 items costing2 or5 each, with total cost31. Is the model feasible for whole items?

x+y10,2x+5y31 givesy11/3,x19/3. This is a unique real solution but not a valid whole-item count; the stated data cannot describe such a box.

Formulae

Systems of Equations Matrix Representation

$$A \mathbf{x} = \mathbf{b} \iff \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$$
Examiner Pitfalls

Common Exam Mistakes in Linear Systems

  • Undefined Variables: Always define variables at the outset: "Let $x$ be the price of..." Omitting variable definitions can lose the method mark $[M1]$.
  • Negative Values in Real Contexts: If your solution gives negative quantities for tickets or weights, check the equations and the feasibility of the original data. A correct algebraic solution can be physically impossible.
Topic 1.5 Check

Exam Mini Test (5 Questions)

Score: 0 / 5

1. Solve $\begin{cases} 2x + y = 10 \\ x - y = 2 \end{cases}$. What is the value of $x$?

2. If two linear equations have identical slopes but different $y$-intercepts, how many solutions exist?

3. Elimination gives0=7. What follows?

4. A ticket system gives2.5adult tickets. What should you conclude?

5. Two proportional equations give0=0. How many real solutions for two unknowns?